Dedekind modules^*

Euclidean modules

“…An ideal I of a commutative ring R is said to be invertible if II = R, where I = {x ∈ K | Ix ⊆ R} and K is the total quotient ring of R. In 1996, Naoum and Al-Alwan [1] generalized this notion to modules. Let M be an R-module, and let S be the set of nonzero divisors of R. One can easily verify that…”

“…Otherwise, the theorem may be false. To prove Theorem 4.5, we need a lemma from [1]. If m 1 = 0, then letting t = 1 and r = r 0 , we obtain 1m = r 0 n. If m 1 = 0, then there exists an r 1 ( = 0) ∈ R such that 0 = r 1 m 1 ∈ Rn by Lemma 5.19 in [5], i.e., there exists an r 2 ∈ R such that r 1 m 1 = r 2 n. Hence r 1 m = r 1 (r 0 n + m 1 ) = r 1 r 0 n + r 2 n = (r 1 r 0 + r 2 )n.…”

“…In 1996, Naoum and Al-Alwan [1] generalized the notion of Dedekind domain to the case of modules and introduced the notion of Dedekind modules. In 2005, Alkan, Sarac, and Tiras [2] gave certain characterizations of Dedekind modules and Dedekind domains.…”

Cancellation Modules and Homogeneous Idealization

“…If N is an invertible submodule of a cancellation module M, then N is cancellation. For if I and J are ideals of R such that IN = JN , then IM = INN −1 = JNN −1 = JM, and hence I = J (see Naoum and Al-Alwan, 1996 for properties of invertible submodules in this sense). Anderson (Anderson, 2001, Theorem 5.3) that if M is a cancellation module over a one-dimensional integral domain R then M is locally cancellation.…”

“…P is cancellation (hence weak cancellation) but P is not join principal (Anderson, 2001). Naoum (1996) introduced and investigated 1 2 (weak) cancellation modules. An R-module M is called 1 2 weak cancellation if for all ideals I of R, M = IM implies R = I + ann M. Obviously, finitely generated modules are 1 2 weak cancellation (Kaplansky, 1974, Theorem 76).…”